1.1 Set-up

Consider integrable Hamiltonian systems with two degrees of freedom (d.o.f.) of the form

(1.1) $$ \begin{align} H(p_x,x,p_{y},y)=\frac{p_x^2}{2}+\frac{p_y^2}{2}+V_1(x)+V_2(y), \end{align} $$

where $(x,y)$ are the space coordinates, $(p_x,p_y)$ are the corresponding momenta and the potentials $V_1,V_2:{\mathbb {R}}\to {\mathbb {R}}_{\geq 0}$ are even unimodal $C^2$ -maps that tend monotonically to infinity with their argument (without loss of generality we take $V_{1}(0)=V_2(0)=0$ : see the precise ‘Deck’ assumption below). The Hamiltonian flow $(\varphi _t)_{t\in {\mathbb {R}}}$ of (1.1) describes a particle which oscillates in a potential well. The Hamiltonian flow on a given energy surface E is foliated by the level sets with fixed partial energies $(E_{1}=H_{1}(I_1),E_{2}=H_{2}(I_2)=E-E_{1})$ , where $I_{i}(E_i)$ is the action of the one-d.o.f. system $H_i$ . For a given energy level $E\geq 0$ and any $0\leq E_{1}\leq E$ let

$$ \begin{align*} S_{E,E_{1}}\kern1.5pt{:=}\kern1.5pt\bigg\{(p_x,x,p_y,y)\kern1.5pt{\in}\kern1.5pt{\mathbb{R}}^4:\frac{p_x^2}{2}+V_1(x)\kern1.5pt{=}\kern1.5ptE_{1},\frac{p_y^2}{2}+V_2(y)\kern1.5pt{=}\kern1.5ptE-E_{1},(x,y)\kern1.5pt{\in}\kern1.5pt {\mathbb{R}}^2\bigg\}. \end{align*} $$

Then the phase space of the flow $(\varphi _t)_{t\in {\mathbb {R}}}$ , is foliated by the invariant sets $\{S_{E,E_{1}}:E\geq 0,0\leq E_{1}\leq E\}$ , which are tori for $0<E_{1}<E,$ and, for $E>0$ and $E_1\in \{0,E\}$ , are circles. Denote the restriction of $(\varphi _t)_{t\in {\mathbb {R}}}$ to $S_{E,E_{1}}$ by $(\varphi ^{E,E_{1}}_t)_{t\in {\mathbb {R}}}$ . The smooth flow without reflection is trivially integrable and oscillatory. The projection of $S_{E,E_{1}}$ to the configuration space is the projected rectangle [Reference Pnueli and Rom-Kedar16]

(1.2) $$ \begin{align} R^{(E,E_1)}=[-x^{\max}(E_{1}),x^{\max}(E_{1})]\times[-y^{\max}(E-E_{1}),y^{\max}(E-E_{1})], \end{align} $$

where $V_{1}(x^{\max }(E_{1}))=E_{1}, V_{2}(y^{\max }(E_{2}))=E_{2}, E_2=E-E_1$ . The union of all iso-energy rectangles is the Hill region: $\mathcal {D}_{\mathrm {Hill}}(E)=\bigcup _{0\leq E_{1}\leq E} R^{(E,E_1)}=\{(x,y)|V_{1}(x)+V_{2}(y)\leqslant E\}$ (see [Reference Pnueli and Rom-Kedar16] for more general formulation).

Denote by $\omega _{i}(E_i)={2\pi }/{T_{i}(E_i)}$ the frequency in each degree of freedom, where $T_{i}(E_i)$ is the period of oscillation. The standard transformation to action angle coordinates $(I_{i},\theta _i)$ in each degree of freedom renders (1.1) into the form $H(p_x,x,p_y,y)=H_{1}(I_{1})+H_2(I_2)$ and in these coordinates the flow is simply

$$ \begin{align*} (\varphi^{E,E_1}_t)&=\big(I_{1}(E_{1}),\theta_1(t)=\omega_{1}(E_1)t+\theta_1(0),\\ & \quad{}I_{2}(E-E_{1}),\theta_2(t)=\omega_{2}(E-E_1)t+\theta_2(0)\big). \end{align*} $$

Recall that $\omega _{i}(I_i)=H_{i}^{\prime }(I_{i})={dE_{i}}/{dI_{i}}$ and that the Hamiltonian is said to satisfy the twist condition if $\det ({\partial ^{2}H}/{\partial I_{i} \partial I_j})=\prod \omega _{i}^{\prime }(I_i)\neq 0$ and to satisfy the iso-energy non-degeneracy condition if

$$ \begin{align*} \begin{vmatrix}\dfrac{\partial^{2}H}{\partial I_{i} \partial I_j} & \dfrac{\partial H}{\partial I_{i} } \\ \dfrac{\partial H}{\partial I_{j} } & 0 \\ \end{vmatrix}= \begin{vmatrix}\omega_{1}^{\prime}(I_1) & 0 & \omega_1(I_1) \\ 0 & \omega_{2}^{\prime}(I_2) & \omega_2(I_2)\\ \omega_1(I_1) & \omega_{2}(I_2) & 0\\ \end{vmatrix}=-\omega_{1}^{\prime}(I_1)\omega_2^{2}(I_2)-\omega_1^{2}(I_1)\omega_{2}^{\prime}(I_2)\neq0. \end{align*} $$

The character of the smooth flow $(\varphi_t^{E,E_1})$ on the level set $S_{E,E_1}$ depends on the frequency ratio on this level set. If

$$ \begin{align*} \Omega=\Omega(E,E_1)=\frac{\omega_{1}(E_1)}{\omega_{2}(E-E_1)}=\frac{H^{\prime}_{1}(I_{1}(E_1))}{H^{\prime}_{2}(I_{2}(E-E_1))} \end{align*} $$

is rational then the flow is periodic (the resonant case), and if $\Omega $ is irrational then it is quasi-periodic (the non-resonant case). Recall that $({d}/{dE_{1}})\Omega (E,E_1)\neq 0$ if and only if the iso-energy non-degeneracy condition is satisfied.

Now, assume that the particle is confined to a bounded polygonal room $P\subset {\mathbb {R}}^2$ whose walls consist of vertical and horizontal segments. When the particle meets the wall it reflects elastically. More precisely, if a trajectory meets a vertical segment at $(p_x,x,p_y,y)$ then it jumps to $(-p_x,x,p_y,y)$ and continues its movement in accordance with the Hamiltonian flow solving

(1.3) $$ \begin{align} \frac{dp_x}{dt}=-V_1^{\prime}(x), \quad\frac{dx}{dt}=p_x,\quad \frac{dp_y}{dt}=-V_2^{\prime}(y), \quad\frac{dy}{dt}=p_y. \end{align} $$

Similarly, if a trajectory meets a horizontal segment at $(p_x,x,p_y,y)$ then it jumps to $(p_x,x,-p_y,y)$ and continues its movement with (1.3); see [Reference Becker, Elliott, Firester, Cohen, Pnueli and Rom-Kedar1] for the general construction, a mechanical example and the description of the resulting dynamics on energy surfaces, and [Reference Pnueli and Rom-Kedar16] for the global structure of energy surfaces of such systems.

In particular, since all the walls are either horizontal or vertical, the partial energies are preserved under these reflections, so the motion remains restricted to level sets:

$$ \begin{align*} S^{P}_{E,E_{1}}:=\{(p_x,x,p_y,y)\in{\mathbb{R}}^4:H_{1}(x,p_x)=E_1,H_{2}(y,p_y)=E-E_{1},(x,y)\in P\}. \end{align*} $$

Denote the restriction of the impact Hamiltonian flow $(\varphi _t)_{t\in {\mathbb {R}}}$ to $S^{P}_{E,E_{1}}$ by $(\varphi ^{P,E,E_{1}}_t)_{t\in {\mathbb {R}}}$ , see, for example, a trajectory segment projected to the configuration space in Figure 1. Notice that if a particle hits any corner of P then it dies. Namely, the flow is fully defined for regular trajectories and is only partially defined on the set which corresponds to all forward and backward images of corner points.

Figure 1 The living space of the particle.

We are interested in studying the topology of the level set $S^{P}_{E,E_{1}}$ and the invariant measures of $(\varphi ^{P,E,E_{1}}_t)_{t\in {\mathbb {R}}}$ when the total and partial energies $E,E_{1}$ vary. In particular, we ask when the flow is uniquely ergodic. Recall that the flow $(\varphi ^{P,E,E_{1}}_t)_{t\in {\mathbb {R}}}$ is uniquely ergodic if (i) each of its orbits is forward or backward infinite and (ii) if there exists a probability measure $\mu _{E,E_1}$ on $S^P_{E,E_1}$ such that, for every continuous map $f:S^P_{E,E_1}\to {\mathbb {R}}$ and any $(p_x,x,p_y,y)\in S^P_{E,E_1}$ for which the $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ -orbit of $(p_x,x,p_y,y)$ is either forward or backward infinite,

$$ \begin{align*} \lim_{T\to\pm\infty}\frac{1}{T}\int_{0}^Tf(\varphi_{ t}(p_x,x,p_y,y))dt=\int_{S^P_{E,E_1}}f\,d\mu_{E,E_1}, \end{align*} $$

where the $+$ (respectively, $-$ ) sign is taken if the $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ -orbit of $(p_x,x,p_y,y)$ is forward (respectively, backward) infinite. In our case the measure $\mu _{E,E_1}$ is equivalent to the Lebesgue measure on $S^P_{E,E_1}$ .

To formally determine the living space, P, of the particle, by following [Reference Frączek, Shi and Ulcigrai8], denote by $\Xi $ the set of sequences $(\overline {x},\overline {y})=(x_i,y_i)_{i=1}^k$ of points in ${\mathbb {R}}^2_{>0}$ such that

$$ \begin{align*} 0<x_1<x_2<\cdots<x_{k-1}<x_k\quad\text{and}\quad 0<y_k<y_{k-1}<\cdots<y_2<y_1. \end{align*} $$

For every $(\overline {x},\overline {y})\in \Xi $ set $k(\overline {x},\overline {y}):=k$ and denote by $P(\overline {x},\overline {y})$ the right-angled staircase polygon on ${\mathbb {R}}^2$ with consecutive vertices:

$$ \begin{align*} (0,0),(0,y_1),(x_1,y_1),(x_1,y_2),\ldots,(x_{k-1},y_{k-1}),(x_{k-1},y_{k}),(x_k,y_k),(x_k,0); \end{align*} $$

see Figure 2.

Figure 2 Staircase polygons $P(\overline {x},\overline {y})$ , $P(-\overline {x},\overline {y})$ , $P(\overline {x},-\overline {y})$ , $P(-\overline {x},-\overline {y})$ .

Denote by $\Gamma $ the four-element group generated by the vertical and the horizontal reflections $\gamma _v, \gamma _h:{\mathbb {R}}^2\to {\mathbb {R}}^2$ . The polygons of the form

$$ \begin{align*} P(\overline{x},\overline{y}), \; P(-\overline{x},\overline{y})\kern1.5pt{=}\kern1.5pt\gamma_v P(\overline{x},\overline{y}),\; P(\overline{x},-\overline{y})\kern1.5pt{=}\kern1.5pt\gamma_h P(\overline{x},\overline{y}),\; P(-\overline{x},-\overline{y})\kern1.5pt{=}\kern1.5pt\gamma_v\circ\gamma_h P(\overline{x},\overline{y}) \end{align*} $$

are called staircase polygons; see Figure 2. The numbers $x_1,\ldots , x_{k-1}$ are called staircase lengths, and $y_2,\ldots , y_k$ are called staircase heights of the staircase polygon $P(\pm \overline {x},\pm \overline {y})$ . The number $x_{k}$ is called the width and $y_1$ is called the height of $P(\pm \overline {x},\pm \overline {y})$ .

We assume that the living space P of the particle is the union of four staircase polygons determined by the four sequences $(\overline {x}^{\varsigma _1\varsigma _2},\overline {y}^{\varsigma _1\varsigma _2})\in \Xi $ , $\varsigma _1,\varsigma _2\in \{\pm \}$ :

$$ \begin{align*} P(\overline{x}^{++},\overline{y}^{++}), \; P(-\overline{x}^{-+},\overline{y}^{-+}),\; P(\overline{x}^{+-},-\overline{y}^{+-}),\; P(-\overline{x}^{--},-\overline{y}^{--}), \end{align*} $$

such that

$$ \begin{align*} y^{++}_1=y^{-+}_1,\ y^{+-}_1=y^{--}_1,\ x^{++}_{k(\overline{x}^{++}, \overline{y}^{++})}=x^{+-}_{k(\overline{x}^{+-},\overline{y}^{+-})},\ x^{-+}_{k(\overline{x}^{-+}, \overline{y}^{-+})}=x^{--}_{k(\overline{x}^{--},\overline{y}^{--})}. \end{align*} $$

The space of all such polygons we denote by $\mathscr {R}$ . It is the set of star-shaped polygons with a kernel that includes the origin and with only vertical and horizontal boundaries. When the sequences $(\overline {x}^{\varsigma _1\varsigma _2},\overline {y}^{\varsigma _1\varsigma _2})$ are identical (independent of $\varsigma _1,\varsigma _2$ ), the polygon is symmetric to reflections. Our analysis applies to both the general and the symmetric cases (Figure 5 shows symmetric polygons whereas all other figures are related to the general case).

The corners $(x^{\varsigma _1\varsigma _2}_{j},y^{\varsigma _1\varsigma _2}_{j})$ are $90^{\circ }$ corners, henceforth called convex corners, and $(x^{\varsigma _1\varsigma _2}_{j},y^{\varsigma _1\varsigma _2}_{j+1})$ are $270^{\circ }$ corner, henceforth called concave corners. The four-dimensional vector $\{k(\overline {x}^{\varsigma _1\varsigma _2},\overline {y}^{\varsigma _1\varsigma _2})\}_{ \varsigma _1,\varsigma _2\in \{\pm \}}$ is called the topological data of the polygon P, whereas the set of four vectors, $\{(\overline {x}^{\varsigma _1\varsigma _2},\overline {y}^{\varsigma _1\varsigma _2}), \varsigma _1,\varsigma _2\in \{\pm \}\}$ , is called the numerical data of the polygonFootnote ^† P.

The level set topology of $S^{P}_{E,E_{1}}$ is determined by the properties of $R^{(E,E_{1})}\cap P$ (see Figure 3) and can be found under mild conditions on the potential.

Theorem 1.2. Assume $V_1,V_2:{\mathbb {R}}\to {\mathbb {R}}_{\geq 0}$ are $C^2$ unimodal potentials (satisfying condition (♢) below). Let P be any polygon in $\mathscr R$ . Then for any $E>0$ , for $E_{1}\in (0,E)$ , the genus of the level set $(E_{1}, E-E_1)$ , $S^{P}_{E,E_{1}}$ , is given by $1$ plus the number of concave corners in $R^{(E,E_{1})}\cap P$ :

$$ \begin{gather*} g(E,E_1)=1+\sum_{\varsigma_1,\varsigma_2\in\{\pm\}}\#\{1\leq k<k(\bar{x}^{\varsigma_1\varsigma_2},\bar{y}^{\varsigma_1\varsigma_2}):V_1(x^{\varsigma_1\varsigma_2}_k)<E_1<E-V_2(y^{\varsigma_1\varsigma_2}_{k+1})\}. \end{gather*} $$

Specifically, for $E> 0$ the interval $E_{1}\in (0,E)$ is divided into a finite number of segments on which the level sets have a constant genus. This partition is non-trivial for $E>{\min } _{\varsigma _1,\varsigma _2,k}V_1(x^{\varsigma _1\varsigma _2}_k)+V_2(y^{\varsigma _1\varsigma _2}_{k+1}) $ . Close to the end points of $(0,E)$ the genus is $1$ , whereas for E sufficiently large there exists an interval of level sets, $E_{1}\in I_{\max }$ with genus $g_{\max }=\sum _{\varsigma _1,\varsigma _2\in \{\pm \}}k(\bar {x}^{\varsigma _1\varsigma _2},\bar {y}^{\varsigma _1\varsigma _2})-3$ and on which for almost all $E_{1}$ values the motion is uniquely ergodic.

Figure 3 A projected rectangle and the staircase polygon. Cyan—the projected rectangle. Yellow—the polygon $R^{(E,E_{1})}\cap P$ includes three concave corners, thus, by Theorem 1.2, $g(E,E_{1})=4$ .

Figures 3 and 4 provide the geometrical interpretation of this theorem in the configuration space, where several iso-energy rectangles are plotted on top of an asymmetric staircase polygon P. Figure 4 demonstrates that the number of concave corners that are included in $R^{(E,E_{1})}\cap P$ can vary at a fixed energy. Figures 5 and 6 demonstrate the division of the intervals $E_{1}\in [0,E]$ of iso-energy level sets into a finite number of intervals, each having level sets with a fixed genus. The figures show how this division depends on the energy E. In these plots, called impact energy-momentum bifurcation diagrams (IEMBDs; see [Reference Pnueli and Rom-Kedar15, Reference Pnueli and Rom-Kedar16]) the regions in the $(E,E_{1}) $ plane at which level sets include impacts with certain parts of the boundaries are shown. Figure 5 shows these plots for symmetric polygons and Figure 6 shows it for the asymmetric polygon of Figure 4. The gray wedge in each IEMBD corresponds to all allowed level sets (since $E_{1}\in [0,E]$ ). A family of iso-energy level sets corresponds to a vertical line in this plot. The projected rectangles of the iso-energy level sets shown in Figure 4 at two energies correspond to the cyan squares on the two vertical black lines of Figure 6. Each blue colored wedge in the IEMBD corresponds to level sets that include impacts with a concave corner of one of the polygons $P(\overline {x}^{\varsigma _{1},\varsigma _{2}}, \overline {y}^{\varsigma _{1},\varsigma _{2}})$ . If j polygons have the same concave corner we say that this wedge has multiplicity j, so in Figure 5 each concave corner has multiplicity 4. The pink (respectively, light-green) regions correspond to level sets that impact the extreme vertical (respectively, horizontal) boundaries of a polygon. Regions that are in the complement to the blue wedges correspond to level sets with genus 1 (so, in particular, small and large ( $E_{1}\approx E$ ) values are included in this set). Regions that are in $ k_g$ shaded blue wedges (counting multiplicities) have genus $ k_g+1$ . Thus, the regions in the intersection of all the blue wedges have the maximal genus $g_{\max }$ . The intersection of this region with the two pink and two light green wedges corresponds to level sets that, for almost all $E_{1}$ , have uniquely ergodic dynamics with the very mild assumption (condition (♢)) on the potentials. As described next, with stronger assumptions on the potentials we prove unique ergodicity for almost all level sets in the allowed region, whereas for quadratic resonant potentials the IEMBD provides a more delicate division to segments as will be explained in §1.4.

Figure 4 The intersection of a star-shaped polygon (gray) with four iso-energy projected rectangles (cyan) at (a) $E=2.7$ and (b) $E=5.7 $ energy values. By Theorem 1.2, the genera of the corresponding iso-energy level sets for the four $R^{(E,E_{1})}$ rectangles shown are, for increasing $E_{1}$ (a) $ g(E=2.7,E_1)=\{1,2,4,1\}$ (b) $ g(E=5.7,E_1)=\{1,2,5=g_{\max },1\}$ . The potentials here are quadratic with $\omega _{1}=1,\omega _{2}=0.8\sqrt {2}$ .

Figure 5 Impact energy-momentum bifurcation diagram: (a) for a symmetric cross (one concave corner with multiplicity $4$ ); (b) symmetric two-step cross (two concave corners each with multiplicity $4$ ). Here, due to symmetry, each blue wedge corresponds to four overlapping wedges, so $R^{(E,E_{1})}\cap P$ includes $4k$ concave corners if and only if $(E,E_{1})$ is in a region covered by $4k$ overlapping shaded blue regions. Only for these regions is the level sets genus greater than $1$ . The pink (respectively, light-green) regions correspond to level sets that impact the extreme vertical (respectively, horizontal) sides.

Figure 6 Impact energy-momentum bifurcation diagram for the asymmetric cross shown in Figure 4 (four distinct concave corners). The colored circles correspond to the energies of the corners of the corresponding colored staircase polygons shown in the inset. All corners have distinct partial energies, so $R^{(E,E_{1})}\cap P$ includes k concave corners if and only if $(E,E_{1})$ is in a region covered by k overlapping blue regions. Only for these regions is the level sets genus greater than $1$ . The pink (respectively, light green) regions include extremal vertical (respectively, horizontal) boundaries. The two vertical lines indicate the energies $E=2.7,5.7 $ and the cyan squares on these lines correspond to the $E_{1}$ values of the rectangles shown in Figure 4.

1.2 Deck potentials

To study the properties of the invariant measures of the flow $(\varphi ^{P,E,E_{1}}_t)_{t\in {\mathbb {R}}}$ we need additional assumptions on the potentials. For all $z_0\in {\mathbb {C}}$ and $r>0$ define the ball centered at $z_0$ and the droplet emanating from $z_0$ by

$$ \begin{align*}B(z_0,r)=\{z\in {\mathbb{C}}:|z-z_0|<r\}\quad\text{and}\quad C(z_0,r)=\bigcup_{s\in(0,1]}sB(z_0,r)\quad\text{if}\;r<|z_0|.\end{align*} $$

We define a special class of even potentials $V_1,V_2$ denoted by $\textit {Deck}$ . An even $C^2$ -map $V:{\mathbb {R}}\to {\mathbb {R}}$ belongs to $\textit {Deck}$ if

(♢) $$ \begin{align} &V(0)=V^{\prime}(0)=0,\quad V^{\prime}(x)>0\quad \text{for all}\ x>0\quad \text{and}\quad \lim_{x\to+\infty}V(x)=+\infty; \end{align} $$

(♡) $$ \begin{align} &V:(0,+\infty)\to(0, +\infty)\ \text{is an analytic map.} \end{align} $$

Then $V:(0,+\infty )\to (0, +\infty )$ has a holomorphic extension $V:U\to {\mathbb {C}}$ on an open neighborhood $U\subset {\mathbb {C}}$ of $(0,+\infty )$ , and we have that:

(♣) $$ \begin{align} &\text{for every } E>0 \text{ there exist } 0<r<E \text{ and a bounded open set } U_{E}\subset U \nonumber\\ &\text{such that } V:U_E\to V(U_E) \text{ is biholomorphic with } C(E,r)\subset V(U_E); \end{align} $$

(♠) $$ \begin{align} \text{there exists}\ C_{E}>0 \ \text{such that} \ \bigg|\frac{V^{\prime\prime}(z)V(z)}{(V^{\prime}(z))^2}\bigg|\leq C_E\quad \text{for all}\ z\in U_E. \end{align} $$

The class $\textit {Deck}$ contains all unimodal analytic maps (that is, satisfying (♢)); see Proposition A.1. Further examples of $\textit {Deck}$ potentials which are not analytic at $0$ (such as $V(x)=|x|^m\exp (-1/|x|)$ ) are presented in Appendix A. The Deck assumption ensures that the period depends analytically on the energy. Additionally, we will most often assume that $V\in \textit {Deck}$ also satisfies

(☺) $$ \begin{align} \frac{V(x)V^{\prime\prime}(x)}{(V^{\prime}(x))^2}\geq \frac{1}{2}\quad \text{for all}\ x>0,\ \text{or} \end{align} $$

(☼)

Condition (☺) is equivalent to V being the square of a convex function, and ensures that the period is a decreasing function of the energy, while condition (☼) means additionally not being a quadratic function, so the period is strictly decreasing with the energy; see Lemmas 4.4 and 4.5. For example, all non-trivial non-quadratic even polynomials with non-negative coefficients are Deck and satisfy (☼), whereas $V(x)=x^{2}-\sqrt {2}x^4+x^6=(x+x^3)^2-(2+\sqrt {2})x^4$ is Deck but does not satisfy (☺) (cf. Proposition A.4).

1.3 Main results for non-quadratic Deck potentials

The main result (Theorem 1.3) says that for every energy level $E>0$ and typical (almost every, a.e.) partial energy $E_{1}\in [0,E]$ the local flow $(\varphi ^{P,E,E_{1}}_t)_{t\in {\mathbb {R}}}$ is uniquely ergodic whenever at least one potential is not a quadratic function or both are quadratic functions and non-resonant (that is, in this latter case $\Omega (E,E_1)$ , which by Remark 1.1 does not depend on E and $E_1$ , is irrational).

Notice that for small energy (that is, satisfying (1.4)), and, in fact, for all the gray areas in the IEMBD figures (Figures 5 and 6), the motion does not impact the polygon walls and the above theorem trivially holds as the motion on most of the tori is of irrational rotation (in the non-quadratic case, $(\alpha )$ , the iso-energy non-degeneracy condition holds since $\omega _{i}^{\prime }(I_i)\geqslant 0,i=1,2$ , and at least for one oscillator the inequality is strict). The non-trivial statement is that even when impacts occur (the pink, light-green and blueish regions in the IEMBD figures), the flow is usually uniquely ergodic. By the definition of unique ergodicity, the theorem tells us that for most level sets time averages are equivalent to phase-space averages for every initial condition on these level set. The complementary set could have periodic and quasi-periodic motion coexisting on the same level set as in [Reference Becker, Elliott, Firester, Cohen, Pnueli and Rom-Kedar1, Reference McMullen13].

1.4 Linear oscillators case

We study separately the case when $V_1$ and $V_2$ are quadratic. This boundary case (in the class of $\textit {Deck}$ potentials satisfying ☺) is significantly different from the general case. In the quadratic case we have $V_1=\Omega ^{2} V_2$ , and we consider rational $\Omega $ so the harmonic motion is resonant.

If the energy level E is low enough then the impacting resonant quadratic flow, $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ , does not reach the boundary and the motion is trivially identical to the resonant periodic linear oscillator motion (the gray area in the IEMBD figures, before any of the blue wedges emerge).

Proposition 1.4. For energies satisfying

(1.4) $$ \begin{align} E\leq{\min}\{V_1(x^{\varsigma_1\varsigma_2}_k)+V_2(y^{\varsigma_1\varsigma_2}_{k+1}):\varsigma_1,\varsigma_2\in\{\pm\},0\leq k\leq k(\bar{x}^{\varsigma_1\varsigma_2},\bar{y}^{\varsigma_1\varsigma_2})\}, \end{align} $$

where $x^{\varsigma _1\varsigma _2}_0=y^{\varsigma _1\varsigma _2}_{k(\bar {x}^{\varsigma _1\varsigma _2},\bar {y}^{\varsigma _1\varsigma _2})+1}=0$ , the restricted Hamiltonian flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ is identical to $(\varphi ^{E,E_1}_t)_{t\in {\mathbb {R}}}$ for all $E_1\in [0,E]$ , and for the impacting resonant quadratic flow it corresponds to periodic motion.

On the other hand, a non-trivial statement, with a proof which is similar to that of Theorem 1.3, is that if the energy level E is high enough so that at least one of the extremal horizontal or vertical boundaries is reached by orbits in $S^P_{E,E_1}$ for all $E_1\in [0,E]$ then $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ is uniquely ergodic for a.e. $E_1$ (energies beyond which the segment $[0,E] $ is covered by the union of the pink and light green wedges in the IEMBD figures, such as $E \approx 5.7 $ in Figure 6; see also the corresponding projected rectangles in Figure 4(b)).

Theorem 1.5. If

(1.5) $$ \begin{align} E \geq{\underset{\varsigma_1,\varsigma_2\in\{\pm\}}\min}V_1(x_{k(\bar{x}^{\varsigma_1\varsigma_2},\bar{y}^{\varsigma_1\varsigma_2})}^{\varsigma_1\varsigma_2}) +{\underset{\varsigma_1,\varsigma_2\in\{\pm\}}\min}V_2(y_1^{\varsigma_1\varsigma_2}) \end{align} $$

then the impacting resonant quadratic flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ is uniquely ergodic for a.e. $E_1\in [0,E]$ .

Studying the case of intermediate E is much more complex and requires new methods. Then the $E_{1}$ interval $[0,E]$ splits into at most countably many intervals so that for every interval I from this partition we have three possible scenarios:

• (ue) the flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ is uniquely ergodic for a.e. $E_1\in I$ ;

• (cp) the flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ is completely periodic for all $E_1\in I$ ;

• (coex) for a.e. $E_1\in I$ the phase space of $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ splits into two completely periodic cylinders and two uniquely ergodic components.

For $E>0$ let $\mathcal {J}_E$ denote a partition (into open intervals) of the $E_{1}$ -interval $[0,E]$ determined by the numbers

$$ \begin{align*}V_1(x^{\varsigma_1\varsigma_2}_k), E-V_2(y^{\varsigma_1\varsigma_2}_k)\quad\text{for all }\varsigma_1,\varsigma_2\in\{\pm\},1\leq k\leq k(\bar{x}^{\varsigma_1\varsigma_2},\bar{y}^{\varsigma_1\varsigma_2}).\end{align*} $$

In the IEMBD figures, $\mathcal {J}_E$ corresponds to the partition of the vertical interval $[0,E]$ by the colored wedges. Now we formulate two results relating to the cases (cp) and (ue). For low energies, there are intervals of $E_{1}$ values for which no impacts occur (these are the gray regions in the IEMBD figures).

Proposition 1.6. Suppose that $I\in \mathcal J_E$ is an interval such that for every $\varsigma _1,\varsigma _2\in \{\pm \}$ there exists $1\leq {l}^{\varsigma _1\varsigma _2} \leq {k}(\bar {x}^{\varsigma _1\varsigma _2},\bar {y}^{\varsigma _1\varsigma _2})$ such that

(1.6) $$ \begin{align} I\subset \bigcap_{\varsigma_1,\varsigma_2\in\{\pm\}}[E-V_2(y^{\varsigma_1\varsigma_2}_{{l}^{\varsigma_1\varsigma_2}}),V_1(x^{\varsigma_1\varsigma_2}_{{l}^{\varsigma_1\varsigma_2}})]. \end{align} $$

Then the impacting resonant quadratic flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ is completely periodic for every $E_1\in I$ .

Denote by $I_{\mathrm {nonimp}}(E)$ the collection of $E_1$ intervals on which no impacts occur:

(1.7) $$ \begin{align} I_{\mathrm{nonimp}}(E)=\bigcup_{\substack{ 1\leq {l}^{++} \leq {k}(\bar{x}^{++},\bar{y}^{++})\\ 1\leq {l}^{+-} \leq {k}(\bar{x}^{+-},\bar{y}^{+-})\\ 1\leq {l}^{-+} \leq {k}(\bar{x}^{-+},\bar{y}^{-+})\\ 1\leq {l}^{--} \leq {k}(\bar{x}^{--},\bar{y}^{--}) }} \bigcap_{\varsigma_1,\varsigma_2\in\{\pm\}}[E-V_2(y^{\varsigma_1\varsigma_2}_{{l}^{\varsigma_1\varsigma_2}}),V_1(x^{\varsigma_1\varsigma_2}_{{l}^{\varsigma_1\varsigma_2}})]. \end{align} $$

For sufficiently large $E $ this set is empty, whereas for sufficiently small $E,\ I_{\mathrm {nonimp}}(E)= [0,E]$ . For intermediate values $I_{\mathrm {nonimp}}(E)$ may be composed of several disjoint intervals (e.g. the gray segments for $E=3 $ in Figure 5).

When at least one of the extremal boundaries is reached (the union of the pink and light green wedges in the IEMBD figures), similar to the general case of Theorem 1.3, we again get unique ergodicity for a.e. $E_{1}$ .

Theorem 1.7. For every energy level $E>0$ and almost every

$$ \begin{align*} E_1\in \Big[0,E-{\underset{\varsigma_1,\varsigma_2\in\{\pm\}}\min}V_2(y_1^{\varsigma_1\varsigma_2})\Big] \cup \Big[{\underset{\varsigma_1,\varsigma_2\in\{\pm\}} \min}V_1(x_{k(\bar{x}^{\varsigma_1\varsigma_2},\bar{y}^{\varsigma_1\varsigma_2})}^{\varsigma_1\varsigma_2}),E\Big] \end{align*} $$

the impacting resonant quadratic flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ is uniquely ergodic.

Now assume that $I\in \mathcal J_E$ is an interval such that impacts occur and not with the extremal boundaries (pure blue regions in the IEMBD figures):

(1.8) $$ \begin{align} I\subset I_{\mathrm{intimp}} =\Big(E-{\underset{\varsigma_1,\varsigma_2\in\{\pm\}}\min}V_2(y_1^{\varsigma_1\varsigma_2}),{\underset{\varsigma_1,\varsigma_2\in\{\pm\}}\min}V_1(x_{k(\bar{x}^{\varsigma_1\varsigma_2},\bar{y}^{\varsigma_1\varsigma_2})}^{\varsigma_1\varsigma_2})\Big)\backslash I_{\mathrm{nonimp}}(E). \end{align} $$

This case presents non-uniform ergodic properties and requires new constructions. Assume that $\Omega =n/m$ with m, n coprime natural numbers. Let $\{\textit {red, green}\}$ be a partition of the $(\varsigma _1,\varsigma _2)$ set $\{++,+-,-+,--\}$ into two-element set so that

(1.9) $$ \begin{align} \textit{green} =(\textit{pair}1, \textit{pair}2)=\left\{ \begin{array}{@{}ll} \{++,+-\}&\text{ if } m \text{ is odd and } n \text{ is even,}\\ \{++,-+\}&\text{ if } m \text{ is even and } n \text{ is odd,}\\ \{++,--\}&\text{ if } m \text{ and } n \text{ are odd.} \end{array}\right. \end{align} $$

Recall that a billiard in a rectangle can be reflected three times so that the billiard flow is conjugated to the directed flow on the torus. By rescaling we can always consider the flow to be in the direction $45^{\circ }$ . Denote the left lower corner of the original rectangle by $(--)$ , the upper left corner by $(-+)$ , the lower right corner $(+-)$ and the upper right corner by $(++)$ (see Figure 7). In the rational situation, when the impacts from the boundary of the polygon P are ignored, the torus is filled with periodic orbits. We will see that the partition corresponds to having two periodic orbits of the scaled torus which connect the $\textit{pair}1$ and $\textit {pair}2$ corners. These colored periodic orbits induce coloring of the staircase non-extremal boundaries of $P(\overline {x}^{\varsigma _1\varsigma _2},\overline {y}^{\varsigma _1\varsigma _2})$ , which we call colored sides (see Figure 1, with coloring induced by taking m odd and n even as in the figures of §7).

Figure 7 The billiard table $\mathbf {P}(E_1)$ , the translation surface $M(E_1)$ and the marked torus ${\mathbb {T}}_{m,n}$ .

Next for every $colour\in \{\textit {green, red}\}$ let

$$ \begin{align*} \delta^{colour}(E_1): =\max_{\substack{\varsigma_1\varsigma_2\in colour\\ 1\leq k< {k}(\bar{x}^{\varsigma_1\varsigma_2},\bar{y}^{\varsigma_1\varsigma_2})\\ V_1(x_k^{\varsigma_1\varsigma_2})<E_1<E-V_2(y_{k+1}^{\varsigma_1\varsigma_2})}} \!\bigg(m\arccos\sqrt{\frac{V_1(x_k^{\varsigma_1\varsigma_2})}{E_1}}+n\arccos\sqrt{\frac{V_2(y_{k+1}^{\varsigma_1\varsigma_2})}{E-E_1}}\bigg). \end{align*} $$

We will see that $\delta ^{colour}(E_1)$ is related to the measure of orbits that impact the colored sides of the polygon (see details in §7). Since $\delta ^{\textit {green}}(E_1)+\delta ^{\textit {red}}(E_1)=\pi $ for at most countably many $E_1\in I$ (see Lemma 7.2), the interval I has a partition into open intervals of two kinds $\mathcal U^+_I$ and $\mathcal U^-_I$ so that

(1.10) $$ \begin{align} \begin{split} \delta^{\textit{green}}(E_1)+\delta^{\textit{red}}(E_1)>\pi&\quad\text{for all } E_1\in J \text{ if } J\in \mathcal U^+_I, \\ \delta^{\textit{green}}(E_1)+\delta^{\textit{red}}(E_1)<\pi&\quad\text{for all } E_1\in J \text{ if } J\in \mathcal U^-_I. \end{split} \end{align} $$

We will see that in the first case, there are no non-impacting orbits, whereas in the second case there are also periodic orbits which do not impact any colored side. In this latter case, the impacting orbits can be divided into two separate sets, the red/green set, consisting of orbits impacting the red/green sides.

The following theorem asserts that in the first case, for almost all $E_{1}$ , all orbits are equidistributed, whereas in the second case, for almost all $E_{1}$ , all orbits that impact the red/green sides are equidistributed among the red/green set.

This result completes the description of invariant measure of the restricted Hamiltonian flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ for every $E>0$ and almost every $E_1\in [0,E]$ , when $V_1$ , $V_2$ are quadratic potentials with $V_1=\Omega ^{2} V_2$ and $\Omega $ is rational. We discuss the non-trivial implications of this theorem on ergodic averages of the impacting resonant quadratic flow in §9.

1.5 Strategy of the proof

In §3, using a standard change of coordinates (as observed in [Reference Becker, Elliott, Firester, Cohen, Pnueli and Rom-Kedar1]), we construct an isomorphism between the restricted Hamiltonian flow $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ and the directional billiard flow in direction $\pm \pi /4,\pm 3\pi /4$ on a polygon $\mathbf P_{E,E_1}\in \mathscr R$ . For every $E>0$ this gives a piecewise analytic curve $E_1\in [0,E]\mapsto \mathbf P_{E,E_1}\in \mathscr R$ of billiard flows on polygons in $\mathscr R$ . More precisely, this curve is analytic on every interval $I\in \mathcal J_E$ . The unique ergodicity problem for curves of this type was recently studied by the first author in [Reference Frączek, Shi and Ulcigrai8]. In fact, a slight modification of [Reference Frączek, Shi and Ulcigrai8, Theorem 4.2] (see Theorem 5.1 below) is applied to curves $[0,E]\ni E_1\mapsto \mathbf P_{E,E_1}\in \mathscr R$ to show unique ergodicity $(\varphi ^{P,E,E_1}_t)_{t\in {\mathbb {R}}}$ for a.e. $E_1\in [0,E]$ whenever at least one potential $V_1$ or $V_2$ is not a quadratic function or both are quadratic functions with $\Omega $ irrational, see Theorem 1.3, and Theorem 5.3 in §5. Theorem 5.1 relies on the analysis of functions indicating the length and height of steps and the width and height of staircase polygons that make up the polygon $\mathbf P_{E,E_1}$ , when the parameter $E_1$ varies. The relevant results involving these functions necessary for applications of Theorem 5.1 are presented and proved in §4.

The case when $V_1$ , $V_2$ are quadratic with $V_1=\Omega ^{2} V_2$ and $\Omega $ is rational needs a more subtle version of Theorem 5.1; this is Theorem 7.8. Recall that any directional billiard flow on any right-angled polygon is isomorphic to the translation flow on a translation surface obtained using a so-called unfolding procedure from the polygon. This leads to the study of analytic curves of translation surfaces and their translation flows in a fixed direction. Theorem 7.8 gives a criterion for unique ergodicity of the translation flow for almost every translation surface lying on such a curve (this theorem is set in an abstract framework to allow other applications). While the idea of the proof of Theorem 7.8 is similar to [Reference Frączek, Shi and Ulcigrai8, Theorem 4.2], it needs more subtle reasoning, as it involves a new type of partition of the translation surface into polygons with sides that can be parallel to the direction of the flow. This is the main innovation in relation to the approach used in Theorem 5.1 (and in [Reference Frączek, Shi and Ulcigrai8]). Another important novelty is the use of so-called distinguished sides of partitions. The key assumption of Theorem 7.8 is that every orbit of the directional flow (finite or half-infinite or double-infinite) hits a distinguished side, and the key construction in proving Theorem 1.8 is of a glued surface for which this assumption holds.